Let A be the unit square, $\{u_k\}$ is the set of all L2-normalized Laplacian eigenfunctions with Dirichlet boundary condition. Is it true that for any open subset V, $C_V = \inf\limits_k \int\limits_V dx |u_k(x)|^2 > 0$ ?

2 Answers
2

My offic mate and I believe this is true. By separation of variables the eigenfunctions are just $Csin(\pi kx)sin(\pi ly)$ for some fixed constant $C > 0$. Using the trig identity $\sin^2(x) = (1-\cos(2x))/2$, we see that

These eigenfunctions are zero on the boundary. However, the problem is about arbitrary boundary conditions in which case the eigenfunctions are more complicated.
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GH from MOApr 8 '11 at 17:36

4

@GH: I usually understand "$L^2$ eigenfunctions with Dirichlet boundary conditions" to mean precisely the functions vanishing on the boundary...
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Willie WongApr 8 '11 at 18:06

Willie, thanks. At any rate, what I suggested is a more interesting problem, I think.
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GH from MOApr 8 '11 at 22:34

Thanks, but it is not complete. The difficult case corresponds to degenerate eigenvalues, for which an eigenfunction is a linear combination of the products of sines.
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Denis GrebenkovApr 9 '11 at 7:14

If we take the usual trigonometric basis, it is indeed true as has been pointed out. However, there is a harder form of the question: Some of the eigenvalues are degenerate. If we allow arbritrary eigenfunctions, does the claim remain true?